Spherical law of cosines

Average Error: 17.1 → 3.9

Time: 37.6s

Precision: binary64

Cost: 64960

Math

\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

↓

\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (*
     (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
     (* (cos phi2) (cos phi1)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + (fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * (cos(phi2) * cos(phi1))))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

↓

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))) * Float64(cos(phi2) * cos(phi1))))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 17.1

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

2. Applied egg-rr 3.9

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]

3. Simplified 3.9

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right) \cdot R \]
   Proof
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 0 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (fma.f64 (Rewrite<= cos-neg_binary64 (cos.f64 (neg.f64 lambda2))) (cos.f64 lambda1) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 10 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 (neg.f64 lambda2)) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 0 points increase in error, 10 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (+.f64 (*.f64 (Rewrite=> cos-neg_binary64 (cos.f64 lambda2)) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 10 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 0 points increase in error, 10 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))) R): 0 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (+.f64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)))))) R): 10 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (+.f64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))))))) R): 0 points increase in error, 10 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))) R): 0 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))))) R): 0 points increase in error, 0 points decrease in error

4. Final simplification 3.9

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R \]

Alternatives

Alternative 1
Error	10.5
Cost	58824

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0004:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \log \left(e^{\cos \phi_2}\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	3.9
Cost	58688

\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]

Alternative 3
Error	10.5
Cost	52552

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00037:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \log \left(e^{\cos \phi_2}\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	10.5
Cost	52424

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \log \left(e^{\cos \phi_2}\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	23.2
Cost	39369

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -7.6 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 600\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 6
Error	23.7
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.043:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\right)\\ \end{array} \]

Alternative 7
Error	17.1
Cost	39232

\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

Alternative 8
Error	29.7
Cost	32972

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.00038:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 13.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 9
Error	33.3
Cost	32840

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 13.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 10
Error	31.6
Cost	32836

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.0085:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 11
Error	29.7
Cost	32708

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.000215:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 13.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 12
Error	39.8
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.1 \cdot 10^{-54}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(0.5 \cdot \left(\phi_1 + \left(\left(\phi_1 + \lambda_1\right) - \lambda_1\right)\right)\right) \cdot \cos \left(0.5 \cdot \left(\left(\phi_1 + \lambda_1\right) + \left(\lambda_1 - \phi_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 13
Error	35.4
Cost	26436

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 13.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 14
Error	40.2
Cost	26244

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000185:\\ \;\;\;\;R \cdot \left(0.5 \cdot \pi - \sin^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 15
Error	40.2
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0195:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 16
Error	43.9
Cost	19520

\[R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \]

Alternative 17
Error	49.7
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \end{array} \]

Alternative 18
Error	52.7
Cost	12992

\[R \cdot \cos^{-1} \cos \lambda_1 \]

Alternative 19
Error	60.0
Cost	6720

\[R \cdot \sqrt{\lambda_1 \cdot \lambda_1} \]

Alternative 20
Error	60.9
Cost	448

\[\left(1 + \lambda_1 \cdot R\right) + -1 \]

Alternative 21
Error	60.7
Cost	192

\[\lambda_1 \cdot R \]

Reproduce

herbie shell --seed 2022340 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))